If the lateral surface of right circular cone is 2 times its base , then the semi-vertical angle of the cone must be :

Options

A. `15^(@)`
B. `30^(@)`
C. `45^(@)`
D. `60^(@)`

Answer

To solve the problem, we need to find the semi-vertical angle of a right circular cone given that the lateral surface area is 2 times the area of its base.

### Step-by-Step Solution:

1. **Understand the Formulas**:
- The lateral surface area (LSA) of a cone is given by the formula:
\[ \text{LSA} = \pi R L \]
where \( R \) is the radius of the base and \( L \) is the slant height.
- The area of the base (A) of the cone is given by:
\[ A = \pi R^2 \]

2. **Set Up the Equation**:
- According to the problem, the lateral surface area is 2 times the area of the base:
\[ \pi R L = 2 \times \pi R^2 \]

3. **Simplify the Equation**:
- We can cancel \( \pi \) from both sides (assuming \( R \neq 0 \)):
\[ R L = 2 R^2 \]
- Now, divide both sides by \( R \) (assuming \( R \neq 0 \)):
\[ L = 2R \]

4. **Relate the Slant Height to the Radius and Height**:
- In a right circular cone, the relationship between the radius \( R \), height \( h \), and slant height \( L \) is given by:
\[ L = \sqrt{R^2 + h^2} \]
- Substituting \( L = 2R \) into the equation:
\[ 2R = \sqrt{R^2 + h^2} \]

5. **Square Both Sides**:
- Squaring both sides to eliminate the square root:
\[ (2R)^2 = R^2 + h^2 \]
\[ 4R^2 = R^2 + h^2 \]

6. **Solve for Height**:
- Rearranging the equation gives:
\[ 4R^2 - R^2 = h^2 \]
\[ 3R^2 = h^2 \]
- Taking the square root:
\[ h = \sqrt{3}R \]

7. **Find the Semi-Vertical Angle**:
- The semi-vertical angle \( \theta \) can be found using the tangent function:
\[ \tan(\theta) = \frac{h}{R} \]
- Substituting \( h = \sqrt{3}R \):
\[ \tan(\theta) = \frac{\sqrt{3}R}{R} = \sqrt{3} \]

8. **Determine the Angle**:
- The angle \( \theta \) for which \( \tan(\theta) = \sqrt{3} \) is:
\[ \theta = 60^\circ \]
- However, we need the semi-vertical angle, which is half of this angle:
\[ \text{Semi-vertical angle} = \frac{60^\circ}{2} = 30^\circ \]

### Final Answer:
The semi-vertical angle of the cone is \( 30^\circ \).